A272210 Difference table of the divisors of the positive integers (with every table read by antidiagonals upwards).

1, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 4, 1, 4, 5, 1, 1, 2, 0, 1, 3, 2, 2, 3, 6, 1, 6, 7, 1, 1, 2, 1, 2, 4, 1, 2, 4, 8, 1, 2, 3, 4, 6, 9, 1, 1, 2, 2, 3, 5, 0, 2, 5, 10, 1, 10, 11, 1, 1, 2, 0, 1, 3, 0, 0, 1, 4, 1, 1, 1, 2, 6, 1, 2, 3, 4, 6, 12, 1, 12, 13, 1, 1, 2, 4, 5, 7, -2, 2, 7, 14, 1, 2, 3, 0, 2, 5, 8, 8, 10, 15

Offset

1,4

Comments

This is an irregular tetrahedron in which T(n,j,k) is the k-th element of the j-th antidiagonal (read upwards) of the difference table of the divisors of n.

The first row of the slice n is also the n-th row of the triangle A027750.

The bottom entry of the slice n is A187202(n).

The number of elements in the n-th slice is A000217(A000005(n)) = A184389(n).

The sum of the elements of the n-th slice is A273103(n).

The antidiagonal sums give A273262.

If n is a power of 2 the diagonals are also the divisors of the powers of 2 from 1 to n, for example if n = 8 the finite sequence of diagonals is [1], [1, 2], [1, 2, 4], [1, 2, 4, 8].

First differs from A273132 at a(89).

Links

Table of n, a(n) for n=1..102.

Example

The tables of the first nine positive integers are
1; 1, 2; 1, 3; 1, 2, 4; 1, 5; 1, 2, 3, 6; 1, 7; 1, 2, 4, 8; 1, 3, 9;
.  1;    2;    1, 2;    4;    1, 1, 3;    6;    1, 2, 4;    2, 6;
.              1;             0, 2;             1, 2;       4;
.                             2;                1;
.
For n = 18 the difference table of the divisors of 18 is
1, 2, 3, 6, 9, 18;
1, 1, 3, 3, 9;
0, 2, 0, 6;
2, -2, 6;
-4, 8;
12;
This table read by antidiagonals upwards gives the finite subsequence [1], [1, 2], [0, 1, 3], [2, 2, 3, 6], [-4, -2, 0, 3, 9], [12, 8, 6, 6, 9, 18].

Mathematica

Table[Table[#[[m - k + 1, k]], {m, Length@ #}, {k, m}] &@ NestWhileList[Differences, Divisors@ n, Length@ # > 1 &], {n, 15}] // Flatten (* Michael De Vlieger, Jun 29 2016 *)

Crossrefs

Cf. A000005, A000217, A027750, A161700, A184389, A187202, A273102, A273103, A273109, A273135, A273132, A273136, A273261, A273262, A273263.

Sequence in context: A362947 A246064 A238157 * A273132 A294859 A336320

Adjacent sequences: A272207 A272208 A272209 * A272211 A272212 A272213

Keyword

sign,tabf

Author

Omar E. Pol, May 18 2016

Status

approved